To rationalize the numerator of the given expression
[tex]\[
\frac{\sqrt[3]{2 x^5}}{3}
\][/tex]
we need to simplify the cube root in the numerator. Let’s break it down step-by-step.
1. Identify the expression inside the cube root: [tex]\(2 x^5\)[/tex].
2. The cube root of a product can be separated into the product of the cube roots:
[tex]\[
\sqrt[3]{2 x^5} = \sqrt[3]{2} \cdot \sqrt[3]{x^5}
\][/tex]
3. Now, we deal with each of these components separately:
- The cube root of 2 remains [tex]\(\sqrt[3]{2}\)[/tex].
- The cube root of [tex]\(x^5\)[/tex] can be written as [tex]\((x^5)^{1/3}\)[/tex].
4. Simplify the exponent part:
[tex]\[
(x^5)^{1/3} = x^{5 \cdot \frac{1}{3}} = x^{5/3}
\][/tex]
5. Now combine the simplified components:
[tex]\[
\sqrt[3]{2 x^5} = \sqrt[3]{2} \cdot x^{5/3}
\][/tex]
6. Substitute back into the original expression:
[tex]\[
\frac{\sqrt[3]{2 x^5}}{3} = \frac{\sqrt[3]{2} \cdot x^{5/3}}{3}
\][/tex]
Therefore, the rationalized form of the numerator in this expression is:
[tex]\[
\frac{2^{1/3} \cdot x^{5/3}}{3}
\][/tex]
So, the final form of the given expression, with the numerator rationalized, is:
[tex]\[
\frac{2^{1/3} \cdot x^{5/3}}{3}
\][/tex]
This expression is now in a simpler, rationalized form.
